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8    LEBESGUE'S CONVERGENCE THEOREMS       133

12.   Let X, Y be two locally compact spaces and TT : X -> Y a proper continuous mapping
(Section 1 2.7, Problem 2). Let p, be a positive measure on X, and let v — 7r(fi) (Section
13.4, Problem 8). Show that v*(g) = ju*(# o TT) for all functions g ;> 0 on Y. (Consider
first the case in which g has compact support, and then use Section 1 2.7, Problem
2(b).) Show that N <= Y is v-negligible if and only if TT'^N) is ju-negligible.

13.   Assume that there exists in I = [0, 1] an increasing sequence (Hn) of nonmeasurable
sets (with respect to Lebesgue measure A) such that \J Hn = I and A*(Hn) = 0 for all n.

n

Show that AVinf (1 - <pHn)\ = 0, but that A*(l - <pH|I) Ğ 1 for all n.

14.   (a)   Let F be a convex function defined on a convex open set A c R" (Section 8.5,
Problem 8) and let/* (1 ^ k ^ n) be n integrable functions on X, such that the map-
ping xi-ğ(/*(x))isfcSfi takes its values in A, and such that the composite function
jc H-> F(/i(*), . . . , fn(x)) is integrable. Show that if p is bounded (1 3.9.1 6) and ju,(X) = 1 ,
then

) £ F( J/x *&..., J/M dp).

(Use the Hahn-Banach theorem applied to the convex set of points

(fi,..., /„, Ğ) E A xR

such that w^>F(f !,...,/„)-)

Consider in particular the case n = 1, F(f) == e*, and deduce the inequality of the
means

tftul2 • • • u%* :£ aii/i + a2M2 + ---- h amwm,

m

where the uk are >0, the afc are ^>0 and V ak = 1 .

fcĞi

(b)   Suppose that fi is bounded and ^(X) = l. Show that if two functions /^O,
g ^0 are integrable and such that^ ^ 1, then (J/rf/x)(J^ d^) ^ 1 (use (a)).

15.   For each finite sequence (Aj) t $ j ħ „ of /x- integrable sets, put

D(A1,...,An) = (   U   Aj)-(   0  Aj).

\li72in      /        Vl^y^n      /

If not all the Aj are ^-negligible , put

o-M(Ai, . . . , An) = ftOXAjL, . . . , An))lfjL(A1 u - • - u A.).
Show that

OV(A!, . . . , An) ^ ^— - E <rM(A, , Aj)

if none of the Aj is /^-negligible. (Remark that a point of D(Ai} . . . , An) belongs to
at least n - I of the sets D(A* , Aj) (/ <j).)

16. If A is Lebesgue measure on the unit interval I = [0, 1] and if $(I) is the set of non-
empty closed subsets of I (Section 3.16, Problem 3), show that the function Ah- >A(A)
is not continuous with respect to the Hausdorff distance on 3KI). X, we have fi*(/) = lim /*?(/).